Slew Constrained Minimum Cost Buffering

ABSTRACT

A buffer insertion technique addresses slew constraints while minimizing buffer cost. The method builds initial solutions for the sinks, each having an associated cost, slew and capacitance. As a solution propagates toward a source, wire capacitance and wire slew arc added to the solution. When a buffer is selected for possible insertion, the slew of the solution is set to zero while the cost of the solution is incremented based on the selected buffer and the capacitance is set to an intrinsic capacitance of the buffer. The solutions of two intersecting wire branches are merged by adding branch capacitances and costs, and selecting the highest branch slew. The solution sets are updated by disregarding solutions which have a slew component greater than a slew constraint, and any solution that is dominated by another solution is eliminated. The solution having the smallest cost is selected as the final solution.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention generally relates to the design of semiconductorchips and integrated circuits, and more particularly to the use ofbuffer insertion to manage timing and electrical requirements in anintegrated circuit design.

2. Description of the Related Art

Integrated circuits are used for a wide variety of electronicapplications, from simple devices such as wristwatches, to the mostcomplex computer systems. A microelectronic integrated circuit (IC) chipcan generally be thought of as a collection of logic cells withelectrical interconnections between the cells, formed on a semiconductorsubstrate (e.g., silicon). An IC may include a very large number ofcells and require complicated connections between the cells. A cell is agroup of one or more circuit elements such as transistors, capacitors,resistors, inductors, and other basic circuit elements grouped toperform a logic function. Cell types include, for example, core cells,scan cells and input/output (I/O) cells. Each of the cells of an IC mayhave one or more pins, each of which in turn may be connected to one ormore other pins of the IC by wires. The wires connecting the pins of theIC are also formed on the surface of the chip. For more complex designs,there are typically at least four distinct layers of conducting mediaavailable for routing, such as a polysilicon layer and three metallayers (metal-1, metal-2, and metal-3). The polysilicon layer, metal-1,metal-2, and metal-3 are all used for vertical and/or horizontalrouting.

An IC chip is fabricated by first conceiving the logical circuitdescription, and then converting that logical description into aphysical description, or geometric layout. timing objectives and fixingelectrical violations for interconnects. Higher relative interconnectresistance forces buffers to be placed closer together to achieveoptimal performance. It is estimated that the number of buffers willrise dramatically, reaching about 15% of the total cell count forintrablock communications for 65 nanometer technology, and close to800,000 buffers required for 50 nanometer technologies. Therefore, boththe complexity and importance of buffer insertion is increasing in aneven faster pace.

Physical synthesis is prominent in the automated design of integratedcircuits such as high performance processors and application specificintegrated circuits (ASICs). Physical synthesis is the process ofconcurrently optimizing placement, timing, power consumption, crosstalkeffects and the like in an integrated circuit design. This comprehensiveapproach helps to eliminate iterations between circuit analysis andplace-and-route. Physical synthesis has the ability to repower gates,insert buffers, clone gates, etc., so the area of logic in the designremains fluid. However, physical synthesis can take days to complete.

During physical synthesis, buffer insertion is called for to eitheroptimize nets for delay or to fix nets due to electrical violations. Onemechanism for performing buffer insertion on a fixed Steiner integratedcircuit topology is known as the van Ginneken algorithm, described inthe article “Buffer Placement in Distributed RC-tree Networks forMinimal Elmore Delay,” L. van Ginneken, IEEE Proceedings (ISCAS) pp.865-868 (1990). Given a fixed Steiner tree topology, the van Ginnekenalgorithm finds the optimal buffer placement on the topology under anElmore delay model for a single buffer type and simple gate delay model.The primary idea of van Ginneken is to choose a set of buffer candidatelocations that lie on the Steiner topology at some uniformly fixeddistance apart. Buffer insertion then proceeds for that particular setof candidates from sink to source. When a candidate location fallswithin a region that is blocked because it is too densely populated withlogic, that location is skipped and a buffer is This process is usuallycarried out using a “netlist,” which is a record of all of the nets, orinterconnections, between the cell pins. A layout typically consists ofa set of planar geometric shapes in several layers. The layout is thenchecked to ensure that it meets all of the design requirements,particularly timing requirements. The result is a set of design filesknown as an intermediate form that describes the layout. The designfiles are then converted into pattern generator files that are used toproduce patterns called masks by an optical or electron beam patterngenerator. During fabrication, these masks are used to pattern a siliconwafer using a sequence of photolithographic steps. The process ofconverting the specifications of an electrical circuit into a layout iscalled the physical design.

Cell placement in semiconductor fabrication involves a determination ofwhere particular cells should optimally (or near-optimally) be locatedon the surface of a integrated circuit device. Due to the large numberof components and the details required by the fabrication process forvery large scale integrated (VLSI) devices, physical design is notpractical without the aid of computers. As a result, most phases ofphysical design extensively use computer-aided design (CAD) tools, andmany phases have already been partially or fully automated. Automationof the physical design process has increased the level of integration,reduced turn around time and enhanced chip performance. Severaldifferent programming languages have been created for electronic designautomation (EDA), including Verilog, VHDL and TDML. A typical EDA systemreceives one or more high level behavioral descriptions of an IC device,and translates this high level design language description into netlistsof various levels of abstraction.

Faster performance and predictability of responses are elements ofinterest in circuit designs. As process technology scales to thesubmicron regime, interconnect delays increasingly dominate gate delays.Consequently, physical design optimization tools such as floorplanning,placement, and routing are becoming more “timing-driven” than theprevious generation of tools. Owing to the tremendous drop in VLSIfeature size, a huge number of buffers (i.e., amplifiers or inverters)are needed for achieving inserted into the next candidate insertionlocation that is not blocked while maintaining the uniform spacing.

As seen in FIG. 1, a Steiner tree representation of a net 1 placescandidate buffer insertion points 2 at regular intervals along the pathsfrom source 3 to sinks 4 and 5. A buffer blockage 6 such as a memoryarray or IP core is present in the path of the net. The portion of thenet block by logic cell 6 is skipped when determining candidate bufferinsertion points 2. The spacing between candidate insertion points maybe increased or decreased by the designer to achieve a particular timingrequirement. Increasing the frequency of buffer insertion locations canimprove timing of the net, but at an increased buffer cost, i.e., anincreased size or area of the silicon chip used for the buffers.

A variation of the van Ginneken approach can be used to minimize a costfunction subject to given timing constraints, as explained in thearticle “Optimal Wire Sizing and Buffer Insertion for Low Power and aGeneralized Delay Model,” J. Lillis et a]. IEEE Journal of Solid StateCircuits, vol. 31, no. 3, pp. 437-447 (1996). According to thatmethodology, different possible solutions are examined at each potentialbuffer node, progressing in a bottom-up direction from a sink to asource (postorder traversal). FIG. 1 illustrates two possible solutionsfor the buffer location next to sink 5, a first solution with no bufferand a second solution with a buffer at that location. The next iterationof the analysis has four possible solutions that build on the first setof solutions: one solution with no buffer at either of the twolocations; two solutions with only one buffer location at either of thetwo locations; and one solution with buffers at each of the twolocations. At any point in the analysis “suboptimal” solutions may beeliminated. Each solution is associated with a three-tuple comprised ofcapacitance/cost/slack (C, W, Q), and a solution is deemed suboptimal ifeach of these three components is worse than the correspondingcomponents of any other solution. In this example, the fourth solutionwith buffers at both of the first two buffer locations is suboptimal andremoved from further consideration.

Two possible solutions are similarly built for the branch starting withsink 4. There are accordingly six possible solutions presented to thenext buffer location upstream from logic cell 6. The process ofpropagating different solutions continues for all wire branches and allbuffer locations until the source (driver) is reached, with suboptimalsolutions continuously pruned. Solutions which violate the delayconstraint at the driver arc also pruned. The set of solutions whichsurvive thus excel in at least one aspect of downstream capacitance,buffer cost or slack (required arrival time, or RAT). The bufferassignments selected as the final solution correspond to the solutionwith the smallest overall cost.

While this approach provides a valuable tool for delay optimization, itdoes not address other important electrical characteristics such asslew. Slew (or slew rate) refers to the rise time or fall time of aswitching digital signal. Different definitions can be used to quantifyslew, the most common being the 10/90 slew which is the time it takesfor a waveform to cross from the 10% signal level to the 90% signallevel. Other definitions such as 20/80 slew or 30/70 slew are often usedwhen the waveform has a slowly rising or falling tail. Since higherinterconnect resistivity also causes signal integrity to degrade morequickly with each advancing technology, buffers need to be inserted onlong interconnects to meet slew constraints. The number of buffers on atypical chip is rising dramatically due to this increasing interconnectresistance. For example, an application-specific integrated circuit(ASIC) designed for use in a computer server might have between 2 and 3million gates. Electrical correction for such a circuit might take aslong as 46 hours, with the insertion of as many as 500,000 buffers. Thislarge number of buffers can degrade overall design performance byforcing the rest of the logic to be spread further apart to accommodatethe buffers. The buffers themselves are a drain on power and can causeother gates to be sized to higher power levels since they are furtherapart on the chip.

Prior art buffer insertion tools fail to adequately deal with slewconstraints as they pertain to the increased costs from buffering. Anextension of the Lillis algorithm accommodates a generalized decay modelwhich takes into account the effect of signal slew of buffer delay whichcan contribute to overall delay. However, that approach merelyintegrates slew constraints while still optimizing for delay, andbuffering of non-critical nets may result in unnecessary runtime andresource overhead. Other approaches have different limitations.Length-based buffering, as described in the article “A PracticalMethodology for Early Buffer and Wire Resource Allocation,” C. Alpert etal., ACM/IEEE Proceedings (DAC) pp. 189-194 (2001), tries to reducebuffer and wire congestion but it can be area inefficient especially inthe presence of blockages and its handling of multi-fanout nets. Anothermethod described in the article “Minimum-Buffered Routing ofNon-Critical Nets for Slew Rate and Reliability Control,” C. Alpert etal., IEEE/CAN Proceedings (ICCAD) pp. 408-415 (2001), addresses slewconstraints without regard to delay; however that technique does notmodel slew, it just simplifies the slew constraint to be equivalent to acapacitance constraint, so interconnected resistivity is not actuallymodeled. While appropriate for vary large fanout nets (e.g., over 1000sinks), it essentially becomes equivalent to length-based buffering. Itwould, therefore, be desirable to devise an improved method of bufferinsertion which could control slew violations while reducing orminimizing buffer cost. It would be further advantageous if the methodcould be carried out independently of the timing analysis.

SUMMARY OF THE INVENTION

It is therefore one object of the present invention to provide animproved method of buffer insertion for a net of an integrated circuitdesign.

It is another object of the present invention to provide such a methodwhich reduces or minimizes the buffer cost (area) while maintaininglegal slew constraints.

It is yet another object of the present invention to provide a methodand system for buffer insertion which is efficient in the presence ofblockages and handling of multi-fanout nets.

The foregoing objects are achieved in a method of providing a buffertree for a net of an integrated circuit design, by building an initialsolution set for the buffer tree from initial solutions for sinks of thenet wherein each solution has an associated cost, slew and capacitance,generating sets of candidate solutions for wire branches which originateat the sinks and extend toward a source wherein the candidate solutionsinclude components from the initial solutions and buffers positionedalong the wire branches, updating the candidate solution sets bydisregarding any candidate solutions which have a slew component greaterthan a slew constraint, and selecting a final solution which has asmallest cost from the updated candidate solution sets. In oneimplementation the buffers are selected from a buffer library havingdifferent buffer types, and candidate solutions are generated for eachbuffer type in the library. The initial solutions for each sink have acost of zero, a slew of zero, and a capacitance corresponding to anintrinsic capacitance of the sink. As a solution propagates toward thesource, the capacitance of an existing solution is increased by adding awire capacitance of the new wire section and the slew is increased byadding a wire slew of the new wire section. When a buffer is selectedfor insertion at a potential location along the wire branch, the slew ofan existing solution for the wire branch is set to zero while the costof the existing solution is incremented based on the selected buffer andthe capacitance of the existing solution is set to an intrinsiccapacitance of the selected buffer. The solutions of two intersectingwire branches are merged by adding branch capacitances and costs, andselecting the highest branch slew. Any candidate solution that isdominated by another solution is eliminated.

The above as well as additional objectives, features, and advantages ofthe present invention will become apparent in the following detailedwritten description.

BRIEF DESCRIPTION OF THE DRAWINGS

The present Invention may be better understood, and its numerousobjects, features, and advantages made apparent to those skilled in theart by referencing the accompanying drawings.

FIG. 1 is a diagram of a Steiner tree for a net of an integrated circuitdesign showing candidate buffer insertion points at regular intervalsalong the paths from a source to several sinks, according to the priorart van Ginneken algorithm;

FIG. 2 is a block diagram of a computer system programmed to carry outcomputer-aided design of an integrated circuit in accordance with oneimplementation of the present invention;

FIG. 3 is a schematic diagram illustrating an example of a circuithaving various nets whose wiring is to be optimized as part of aphysical synthesis process;

FIG. 4 is a chart illustrating the logical flow for a buffer assignmentprocess in accordance with one implementation of the present invention;

FIG. 5 is a chart illustrating the logical flow for one implementationof a solution set update procedure that is used with the process of FIG.4; and

FIGS. 6A-6D are schematics diagrams of a wire branch showing aprogression for candidate buffer solutions in accordance with oneexample of the present invention.

The use of the same reference symbols in different drawings indicatessimilar or identical items.

DESCRIPTION OF THE PREFERRED EMBODIMENT(S)

The present invention provides a novel method for determining bufferinsertion locations in a net of an integrated circuit design, and isgenerally applicable to any type of IC design, such as general-purposemicroprocessors, memory units or special-purpose circuitry. The methodmay be implemented as part of a physical synthesis process whichoptimizes placement, timing, power consumption, crosstalk effects orother design parameters. As explained more fully below, an exemplaryembodiment of the present invention provides a fast technique which canhandle a large volume of nets to optimally solve slew buffering whilealso reducing buffering cost.

With reference now to the figures, and in particular with reference toFIG. 2, there is depicted one embodiment 10 of a computer systemprogrammed to carry out the buffer insertion in accordance with oneimplementation of the present invention. System includes a centralprocessing unit (CPU) 12 which carries out program instructions,firmware or read-only memory (ROM) 14 which stores the system's basicinput/output logic, and a dynamic random access memory (DRAM) 16 whichtemporarily stores program instructions and operand data used by CPU 12.CPU 12, ROM 14 and DRAM 16 are all connected to a system bus 18. Theremay be additional structures in the memory hierarchy which are notdepicted, such as on-board (L1) and second-level (L2) caches. In highperformance implementations, system 10 may include multiple CPUs and adistributed system memory.

CPU 12, ROM 14 and DRAM 16 are coupled to a peripheral componentinterconnect (PCI) local bus 20 using a PCI host bridge 22. PCI hostbridge 22 provides a low latency path through which processor 12 mayaccess PCI devices mapped anywhere within bus memory or I/O addressspaces. PCI host bridge 22 also provides a high bandwidth path to allowthe PCI devices to access DRAM 16. Attached to PCI local bus 20 are alocal area network (LAN) adapter 24, a small computer system interface(SCSI) adapter 26, an expansion bus bridge 28, an audio adapter 30, anda graphics adapter 32. LAN adapter 24 may be used to connect computersystem 10 to an external computer network 34, such as the Internet. Asmall computer system interface (SCSI) adapter 26 is used to controlhigh-speed SCSI disk drive 36. Disk drive 36 stores the programinstructions and data in a more permanent state including the programwhich embodies the present invention as explained further below.Expansion bus bridge 28 is used to couple an industry standardarchitecture (ISA) expansion bus 38 to PCI local bus 20. As shown,several user input devices are connected to ISA bus 38, including akeyboard 40, a microphone 42, and a graphical pointing device (mouse)44. Other devices may also be attached to ISA bus 38, such as a CD-ROMdrive 46. Audio adapter 30 controls audio output to a speaker 48, andgraphics adapter 32 controls visual output to a display monitor 50, toallow the user to carry out the buffer insertion as taught herein.

While the illustrative implementation provides the program instructionsembodying the present invention on disk drive 36, those skilled in theart will appreciate that the invention can be embodied in a programproduct utilizing other computer-readable media, including transmissionmedia. The program instructions may be written in the C++ programminglanguage for an AIX environment. Computer system 10 carries out programinstructions for an interconnect optimization process that uses novelbuffer insertion techniques to manage timing requirements and electricalviolations. Accordingly, a program embodying the invention may includeconventional aspects of various placement and timing tools, and thesedetails will become apparent to those skilled in the art upon referenceto this disclosure.

The present invention provides an improved method of determining bufferinsertion locations which may be used to optimize slew and buffer costof a net. The invention may be understood with reference to thegeneralized circuit 60 depicted in FIG. 3. Circuit 60 has four drivingelements or primary inputs 62 a, 62 b, 62 c, 62 d, and four output nodesor sinks 64 a, 64 b, 64 c, 64 d. The sources are interconnected to thesinks via gates 66 a, 66 b, 66 c or other combinational logic 68. Thegates 66 and logic 68 act as sinks for upstream nets, and act as sourcesfor downstream nets, forming a total of nine nets. The layout shown inFIG. 3 is one example of how a placement tool might provide wiringbetween the input and outputs based on a netlist. Candidate bufferlocations are established for a Steiner topology of the nets at someuniformly fixed distance apart, such as 250 μm. A finer spacing willprovide better timing but increases analysis time. The invention may beused with a buffer library have various types of buffers, includingsmaller buffers 70 a or larger buffers 70 b. The following nomenclatureis used to identify the various aspects of the net and its buffersolutions;

T—a routing tree of the net, T=(V, E);

V—the set of vertices (branch points v) in the routing tree;

E—the set of edges (wire sections e) in the routing tree;

C—capacitance;

W—buffer cost (area);

S—slew;

B—a buffer library;

b—a specific buffer (b₀=no buffer);

γ—a buffer assignment, or candidate solution for a buffer assignment;

Γ—a solution set;

α—a slew constraint.

The present invention introduces slew into the buffer assignmentalgorithm to collect delay information for slew rate computation so asto enable the invention to perform in a dynamic programming framework. Agiven buffer solution γ is represented by an associated three-tuple (C,S, W) where C denotes the downstream capacitance at the current node, Sdenotes the cumulative slew along a bottom-up computation, and W denotesthe total cost of the solution, i.e., buffer area. An initial solutionset is built by providing initial solutions γ_(s) for each sink of agiven net, where the initial slew and cost are set to zero, and thecapacitance is set to the intrinsic capacitance of the sink. A finalsolution for a wire branch propagates from one or more sinks toward anupstream source (postorder traversal). For example, in FIG. 3 the nethaving a source at gate 66 a will have a solution that propagates fromsink 64 a and gate 66 b toward gate 66 a. As the solution propagatesupstream, capacitance is increased by adding the capacitance of eachwire section, and slew is increased by adding the slew of each wiresection, which may be expressed according to Bakoglu's metric in termsof the Elmore delay. The invention generates sets of new solutions foreach allowable buffer insertion location in respective wire branches,setting the new slew to zero when a buffer is inserted, incrementing thenew cost based on the particular buffer selected, and setting the newcapacitance to the intrinsic capacitance of the buffer. Solution setsfor intersecting branches are merged by adding the branch capacitancesand costs, and selecting the highest branch slew. Solutions arecontinually optimized with regard to the slew constraint, i.e., anycandidate solutions which have a slew component greater than the slewconstraint are disregarded, and dominated solutions are eliminated. Whenthe progression reaches a source (e.g., gate 66 a), the solution withthe least cost is selected as the final solution for that net. Thebuffer insertion algorithm is repeated for each net in circuit 60.

The present invention may be understood in further detail with referenceto the chart of FIG. 4, which illustrates the logical flow of a slewconstrained minimum cost buffering process for a binary routing tree Tin accordance with one implementation. The process begins (80) bybuilding an initial solution set for each sink s where S(γ_(s))=0,W(γ_(s))=0, and C(γ_(s))=C_(s) (the sink capacitance). Each branchpoint/driver v_(t) is iteratively examined (82) in the order given by apostorder traversal of T, and a wire e in one of the branches is furtherselected for analysis (84). Each candidate solution γ for the selectedwire is updated (86) by setting C(γ)=C(γ)+C_(e) and S(γ)=S(γ)+1n9·D_(e),where C_(e) is the edge capacitance and D_(e) is the Elmore delay forthe current wire section (88). The Elmore delay can further be expressedas D_(e)=R_(e)(C_(e)/2+C(γ)) where R_(e) is the lumped resistance of thewire section. The solution set Γ′ corresponding to the branches T′ ofthe selected branch point/driver v_(t) is then updated (assuming nobuffer insertion at the current location, b=b₀) to check whether anysolutions violate the slew constraint and to see if any solutions aredominated (90). The solution set update procedure is described furtherbelow in conjunction with FIG. 5.

After updating the branch solution set, a determination is made as towhether a buffer is allowed at the current position (92). The positionmay be blocked by some design constraint such as a logic cell or gate.If a buffer is allowed, the process continues by selecting one of theavailable buffers b_(i) in the buffer library (94). For each branchsolution (96), a new solution γ′ is generated by setting C(γ′)=C_(bi),S(γ′)=0, and W(γ′)=W(γ)+W_(bi), where C_(bi) is the intrinsiccapacitance of the selected buffer and W_(bi) is the cost of the buffer(98). The branch solution set is again updated with b=b_(i) to checkwhether any new branch solutions violate the slew constraint and to seeif any new branch solutions are dominated (100). New solutions are sogenerated and updated for each branch solution (102), and for eachbuffer type in the buffer library (104).

If no buffer is allowed at the current position, the buffer analysis isskipped and the process continues with merging of the two branches ofthe selected vertex ( 106) by first setting a merged solution set to anempty set, i.e., Γ

=Ø. Each potential pair of branch solutions (one solution from eachbranch, i.e., each γ₁εΓ₁ and γ₂εΓ₂) are selected for analysis (108). Anew merged solution γ′ is generated by setting C(γ′)=C(γ₁)+C(γ₂),W(γ′)=W(γ₁)+W(γ₂), and S(γ′)=max{S(γ₁),S(γ₂)} (110). The merged solutionset is updated with b=b₀ to check whether any new merged solutionsviolate the slew constraint and to see if any new merged solutions aredominated (112). The process continues iteratively at step 108 for eachpair of branch solutions (114). If there are more candidate solutions(116), the process continues iteratively at stop 86. Once all candidatesolutions have been analyzed, the process continues iteratively at step84 for other wire branches (118). Once all the wire branches for theselected vertex have been analyzed, the process continues iteratively atstep 82 for other vertices (120). Once all vertices have been analyzed,infeasible solutions at the driver are eliminated (122), and thesolution with the smallest cost is returned as the final solution (124).

Details of the solution set update procedure are shown in the flow chartof FIG. 5. The procedure receives as inputs a candidate solution γ′, anexisting solution set Γ, a buffer type b, and a slew constraint α. Acheck is made to see whether the new solution violates the slewconstraint, but the buffer type is first examined to see if any bufferis present (130). If no buffer is present at the current location(b=b₀), the slew constraint is simply compared to S(γ′) (132). If theslew for the new solution is greater than the constraint, the procedurereturns the existing solution set without modification (134). If abuffer is present (136), the slew formula for the new solution iscalculated as the root-mean square of the gate slew and the interconnectslew as taught in U.S. Pat. No. 6,868,533, i.e.,

Totalslew=√{square root over ((GateSlew)²+(WireSlew)²)}{square root over((GateSlew)²+(WireSlew)²)}=[(R _(bi) ·C(γ′)+K _(bi))² +S(γ′) ²]^(1/2),

where R_(bi) and K_(bi) are empirical fitting parameters for thespecific buffer type. If this slew calculation is greater than theconstraint, the procedure again returns the existing solution setwithout modification (134).

If the slew for the new solution passes the slew constraint, an existingsolution is selected for domination checks (138). The old solution isconsidered to dominate the new solution if the slew, cost andcapacitance of the old solution are less than or equal to the slew, costand capacitance of the new solution, i.e., if C(γ)≦C(γ′), S(γ)≦S(γ′) andW(γ)≦W(γ′). If the old solution dominates (140), the procedure returnsthe existing solution set without modification (134). If the oldsolution does not dominate, the procedure checks to see if the newsolution dominates (142). The new solution is considered to dominate theold solution if the slew, cost and capacitance of the new solution areless than or equal to the slew, cost and capacitance of the oldsolution, i.e., if C(γ′)≦C(γ), S(γ′)≦S(γ) and W(γ′)≦W(γ). If the newsolution dominates, the old solution is removed from the solution set(144). After the domination checks, the next existing solution isselected for analysis (146). If all existing solutions have beenexamined regarding domination and the new solution has not beeneliminated, the new solution is inserted into the solution set (148),and the procedure returns the modified solution set (134).

FIGS. 6A-6D illustrate an example of how the invention is applied to awire branch 150 having three wire sections 152, 154, 156 between aninput source 158 and a sink 160, with two potential buffer insertionlocations. In this example, the initial solution for sink 160 is givenas (20, 0, 0), that is, a capacitance of 20, a slew of zero 7 and a costof zero, as shown in FIG. 6A. These values represent theoreticaldesigner units that correspond to actual measurements, but the specificunits may vary according to designer preferences. In a typicalimplementation the delay and slew values would correspond to ameasurement on the order of picoseconds, and the capacitance valueswould correspond to a measurement on the order of femtofarads.

As further seen in FIG. 6B, the first wire section 152 has an intrinsiccapacitance of 10 and a delay of 150, and one buffer type is providedfor possible insertion at the first buffer location (in the direction ofpostorder traversal) with that buffer having a capacitance of 5 and adelay of 30. Two new solutions are derived using the foregoing formulas,one for the buffer inserted at the first location and another for nobuffer inserted at that location. If the butter is inserted, thecapacitance becomes five, the slew stays at 0, and the cost is 1. If nobuffer is inserted at the first location, the capacitance becomes 30,the slew 330, and the cost 0. These two solutions propagate towardsource 158 as further shown in FIG. 6C. The second wire section 154 hasa capacitance of 15, and a delay of 200 if no buffer was inserted at thefirst location, or a delay of 120 if a buffer was inserted. The bufferagain has a capacitance of 5, and a delay of 50 if the previous locationhas no buffer, or a delay of 30 if the previous located is buffered.Four new solutions are then derived: one solution has no buffers ateither location and results in a capacitance of 45, a slew of 770, andzero cost; another solution has a buffer at the second location but nobuffer at the first location and results in a capacitance of 5, zeroslew, and a cost of 1, another solution has a buffer at the firstlocation but no buffer at the second location and results in acapacitance of 20, a slew of 264, and a cost of 1; the fourth solutionhas buffers at both locations and results in a capacitance of 5, zeroslew, and a cost of 2.

At this point in the process some solutions might be eliminateddepending upon the slew constraint. For example, if the slew limit is500, then the first of these solutions is eliminated—the final slew whenfactoring in the gate slew of 200 from source 158 is 796. The solution(5, 0, 2) is also eliminated since it is dominated by solution (5, 0,1). Alternatively, if the slew limit is 1000 the latter two solutionswill be eliminated since they are dominated, i.e., solution (20, 264, 1)and solution (5, 0, 2) are both dominated by solution (5, 0, 1). Thisscenario is illustrated in FIG. 6D, which also shows the finalpropagation of the solutions to source 158. Assuming a capacitance of 10for the third wire section 156, two solutions are derived: one solutionhas no buffers at either location and results in a capacitance of 55, aslew of 1430, and zero cost; another solution has a buffer at the secondlocation but no buffer at the first location and results in acapacitance of 15, a slew of 198, and a cost of 1. The first of thesesolutions violates the slew limit and is eliminated, leaving the finalsolution of (15, 198, 1).

The buffering technique of the present invention may be used as part ofa physical synthesis methodology which inserts buffers early in theprocess for electrical correction so that timing analysis uses legalslew constraints. Buffers on critical nets can later be removed andreplaced. It is estimated that only a small fraction (5-10%) of thebuffers in an IC net will need to be re-buffered for delay optimization,as most of the buffers derived from slew-based insertion are sufficientto meet the net's timing criteria. The invention is particularlybeneficial since the designer does not need to know the required arrivaltimes at sinks, so it can be used earlier in the design flow thantraditional buffering techniques. The invention may be performed totallyindependent of the timing analysis, i.e., incremental timing is notrequired between buffering of individual nets.

For a single buffer type, an optimal linear solution is achievable andfor multiple buffer types the present invention still produces anefficient solution. In an experimental computation of CPU requirementsusing the same buffer library, runtime was speeded up by a factor of25-30, with up to a 21% buffer area reduction. In another experimentalcomputation for area comparison with similar runtimes, buffer area wasreduced by 41%-56% (timing buffering used four buffers and slew-basedbuffering uses 48 buffers, with runtimes of about 50 seconds). Thus, thepresent invention not only saves turn-around time, but also results in abuffered circuit that is smaller and consumes less power. The inventionis especially efficient in the presence of blockages and its handling ofmulti-fanout nets.

Although the invention has been described with reference to specificembodiments, this description is not meant to be construed in a limitingsense. Various modifications of the disclosed embodiments, as well asalternative embodiments of the invention, will become apparent topersons skilled in the art upon reference to the description of theinvention. For example, while the present invention has been disclosedin the context of a binary routing tree having only two branches at eachvertex, it could easily he expanded to other tree structures. It istherefore contemplated that such modifications can be made withoutdeparting from the spirit or scope of the present invention as definedin the appended claims.

1. A method of providing a buffer tree for a net of an integratedcircuit design, comprising: building an initial solution set for thebuffer tree from initial solutions for sinks of the net whereinsolutions have an associated cost, an associated slew and an associatedcapacitance; generating sets of candidate solutions for wire branches ofthe net which originate at the sinks and extend toward a source of thenet wherein the candidate solutions include components from the initialsolutions and from one or more buffers positioned along the wirebranches; updating the candidate solution sets by disregarding anycandidate solutions which have a slew component greater than a slewconstraint; and selecting a final solution which has a smallest costfrom updated candidate solution sets.
 2. The method of claim 1 wherein:the buffers are selected from a buffer library having different buffertypes; and the candidate solutions are generated for more than onebuffer type in the library.
 3. The method of claim 1 wherein saidupdating includes eliminating any candidate solution whose associatedcost, slew and capacitance are dominated by the associated cost, slewand capacitance of another candidate solution.
 4. The method of claim 1wherein said generating includes merging solutions of two intersectingwire branches by adding branch capacitances and costs, and selecting ahighest branch slew.
 5. The method of claim 1 wherein a candidatesolution is generated by: increasing the capacitance of an existingsolution for the wire branch by adding a wire capacitance of a new wiresection; and increasing the slew of the existing solution by adding awire slew of the new wire section.
 6. The method of claim 1 wherein acandidate solution is generated by: selecting a buffer for insertion ata buffer insertion location along the wire branch; setting the slew ofan existing solution for the wire branch to zero; incrementing the costof the existing solution based on the selected butler; and setting thecapacitance of the existing solution to an intrinsic capacitance of theselected buffer.
 7. A computer system comprising: one or more processorswhich process program instructions; a memory device connected to saidone or more processors; and program instructions residing in said memorydevice for providing a buffer tree for a net of an integrated circuitdesign by building an initial solution set for the buffer tree frominitial solutions for sinks of the net wherein solutions have anassociated cost, an associated slew and an associated capacitance,generating sets of candidate solutions for wire branches of the netwhich originate at the sinks and extend toward a source of the netwherein the candidate solutions include components from the initialsolutions and from one or more buffers positioned along the wirebranches, updating the candidate solution sets by disregarding anycandidate solutions which have a slew component greater than a slewconstraint, and selecting a final solution which has a smallest costfrom updated candidate solution sets.
 8. The computer system of claim 7wherein; the buffers are selected from a buffer library having differentbuffer types; and the candidate solutions are generated for more thanone buffer type in the library.
 9. The computer system of claim 7wherein the updating of the candidate solutions includes eliminating anycandidate solution whose associated cost, slew and capacitance aredominated by the associated cost, slew and capacitance of anothercandidate solution.
 10. The computer system of claim 7 wherein thegenerating of the sets of candidate solutions includes merging solutionsof two intersecting wire branches by adding branch capacitances andcosts, and selecting a highest branch slew.
 11. The computer system ofclaim 7 wherein a candidate solution is generated by: increasing thecapacitance of an existing solution for the wire branch by adding a wirecapacitance of a new wire section; and increasing the slew of theexisting solution by adding a wire slew of the new wire section.
 12. Thecomputer system of claim 7 wherein a candidate solution is generated by:selecting a buffer for insertion at a buffer insertion location alongthe wire branch; setting the slew of an existing solution for the wirebranch to zero; incrementing the cost of the existing solution based onthe selected buffer; and setting the capacitance of the existingsolution to an intrinsic capacitance of the selected buffer.
 13. Thecomputer system of claim 7 wherein the initial solutions for sinks havea cost of zero, a slew of zero, and a capacitance corresponding to anintrinsic capacitance of the sink.
 14. A computer program productcomprising: a computer-readable medium; and program instructionsresiding in said medium for providing a buffer tree for a net of anintegrated circuit design by building an initial solution set for thebuffer tree from initial solutions for sinks of the net whereinsolutions have an associated cost, an associated slew and an associatedcapacitance, generating sets of candidate solutions for wire branches ofthe net which originate at the sinks and extend toward a source of thenet wherein the candidate solutions include components from the initialsolutions and from one or more buffers positioned along the wirebranches, updating the candidate solution sets by disregarding anycandidate solutions which have a slew component greater than a slewconstraint, and selecting a final solution which has a smallest costfrom updated candidate solution sets.
 15. The computer program productof claim 14 wherein: the buffers are selected from a bulkier libraryhaving different buffer types; and the candidate solutions are generatedfor more than one buffer type in the library.
 16. The computer programproduct of claim 14 wherein the updating of the candidate solutionsincludes eliminating any candidate solution whose associated cost, slewand capacitance are dominated by the associated cost, slew andcapacitance of another candidate solution.
 17. The computer programproduct of claim 14 wherein the generating of the sets of candidatesolutions includes merging solutions of two intersecting wire branchesby adding branch capacitances and costs, and selecting a highest branchslew.
 18. The computer program product of claim 14 wherein a candidatesolution is generated by: increasing the capacitance of an existingsolution for the wire branch by adding a wire capacitance of a new wiresection; and increasing the slew of the existing solution by adding awire slew of the new wire section.
 19. The computer program product ofclaim 14 wherein a candidate solution is generated by: selecting abuffer for insertion at a buffer insertion location along the wirebranch; setting the slew of an existing solution for the wire branch tozero; incrementing the cost of the existing solution based on theselected buffer; and setting the capacitance of the existing solution toan intrinsic capacitance of the selected buffer.
 20. The computerprogram product of claim 14 wherein the initial solutions for sinks havea cost of zero, a slew of zero, and a capacitance corresponding to anintrinsic capacitance of the sink.